Undulation mode

Damping rate of the undulation mode We write the energy density in a more general form to include volume dilatation d ... 

Fig. 5.3.9. Damping rate of the undulation mode in the smectic A phase of CBOOA determined by laser beat spectroscopy for two different sample thicknesses, 200 and 800 nm. Solid lines represent the theoretical curves calculated from (5.3.40) and (5.3.41). (After Ribotta, Salin and Durand. )...

As far as the highly damped undulation modes are concerned, the volume dilatation can justifiably be neglected and the isothermal approximation is probably satisfactory. The equation of motion then reduces to... 

This surprising result prompted Mazenko, Ramaswamy and Toner to examine the anharmonic fluctuation effects in the hydrodynamics of smectics. We have already shown that the undulation modes are purely dissipative with a relaxation rate given by (5.3.39). To calculate the effect of these slow, thermally excited modes on the viscosities, we recall that a distortion u results in a force normal to the layers given by (5.3.32). This is the divergence of a stress, which, from (5.3.53), contains the non-linear term 0,(Vj uf. Thus, there is a non-linear contribution (Vj uf to the stress. Now the viscosity at frequency co is the Fourier transform of a stress autocorrelation function, so that At (co), the contribution of the undulations to the viscosity, can be evaluated. It was shown by Mazenko et that Atj(co) 1 /co. In other words, the damping of first and second sounds in smectics, which should go as >/(oo)oo , will now vary linearly as co at low frequencies. 

Again, as in S, the undulation mode (q = 0) may be expected to make an important contribution to the scattering. 

It is important to note that the lamellar phase is thus stabilized by the balance of a negative interfacial tension (of the free oil/water interface covered by an amphiphilic monolayer), which tends to increase the internal area, and a repulsive interaction between interfaces. The result, Eq. (48), indicates that the scattering intensity in a lamellar phase, with wave vector q parallel to the membranes, should have a peak at nonzero q for d > d due to the negative coefficient of the q term in the spectrum of Eq. (40). just as in the microemulsion phase. This effect should be very small for strongly swollen lamellar phases (in coexistence with excess oil and excess water), as both very small [96]. Very similar behavior has been observed in smectic liquid crystals (Helfrich-Hurault effect) [122]. Experimentally, the lamellar phase under an external tension can be studied with the surface-force apparatus [123,124] simultaneous scattering experiments have to be performed to detect the undulation modes. 

We want to mention parenthetically that the characteristic shape of the one-phase region of the lamellar phase in the Cio E5-water-octane phase diagram can also be understood from a calculation based on an interfacial model of membranes [174]. In this model, the undulation modes are responsible for the swelling of the lamellar phase a disordered phase (which is modeled as a dilute droplet phase for simplicity) competes with the symmetric lamellar phase in the lower part of the phase triangle. 

At leading (quadratic) order in h, the three tension-like quantities, Tframe. Uuo and Fq, thus have identical values. Nevertheless, they might differ from each other due to nonlinear corrections [90, 164—167]. For instance, the bare tension Fq is expected to deviate from the frame tension Fframe due to the effect of fluctuations. The exact value of the correction depends on the ensemble and differs for systems with a fluctuating number of lipids (variable number of undulating modes) or a fixed number of lipids (fixed number of modes). The former case was analyzed by Cai et al. [165] and the latter case by Farago and Pincus [168] and subsequently by a number of other authors [169-171]. Interestingly, the correction has an additive component in both cases. Hence a stress-free membrane has a finite bare tension. 

Figure 4.30 Deformation of amphiphilic membranes (a), (b) thermal fluctuation deformation modes and (c) local steric deformation due to protrusion of molecules, where (a) is termed the undulation mode and (b) the peristaltic mode. [Adapted from J. N. Israelachvili, Intermolecular and Surface Forces, 2nd Edition, Academic Press, London (1991)]...

For q perpendicular to the director, Bq, the second sound mode again disappears, becoming a shear mode and a layer undulation mode. In this case the mode plays the same role in the SmA phase as the director Mode 1 does in nematic liquid crystals. 

As (0 increases towards O), then PD diverges markedly [164], and this differentiates between the two regimes (i.e., A >0 or <0). So at low frequencies for A <0 the undulation modes are still observed, but at higher fields the conductivity mode dominates and again leads to a scattering texture. 

Expeimental propagation of waves in soap films has been investigated by Vrij and coworkers in a series of papers. The undulation mode was foimd to be underdamped, whereas the overdamped peristaltic mode (Figure 4.21c)... 

However, unlike an open membrane, a spherical shell or tethered vesicle cannot bend without being stretched. That is, there is a linear coupling between the out-of-plane undulation modes and the in-plane phonon modes which causes a strong suppression of out-of-plane fluctuations at long... 

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